Tensor Product Between Two Vectors

The tensor product (or the dyadic product) $\otimes$ between two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as

$$(\mathbf{a}\otimes \mathbf{b})\mathbf{c} = (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\ \ \mathbf{c}.$$ (2.8.2.1)

Thus $(\mathbf{a}\otimes \mathbf{b})$ transforms a vector $\mathbf{c}$ into a vector parallel to $\mathbf{a}$. Since it transforms a vector into a vector and obeys (2.8.1.1), it is a linear transformation. Note that

$$\mathbf{a}\otimes \mathbf{b}\ne \mathbf{b}\otimes \mathbf{a}.$$ (2.8.2.2)